3. Equation of Motion

The purpose of this section is twofold: i) to set up the Hamiltonian formalism in preparation of the quantization procedure, and ii) to show that the equations of motion of the shell are equivalent to the matching condition (1). This section is supplemented by Appendix A in which we connect our canonical formalism to the FGG-approach.

Presently, our first step is to define the conjugate momenta corresponding to the dynamical variables N and R:

  

$$\frac{\partial L}{\partial \dot{N}} = 0$$ P_N = (35)

$$\frac{\partial L}{\partial \dot{R}} = - \frac{R}{G_N} \left[ \tanh ^{-1} \left( \frac{\dot{R}}{\beta} \right) \right \rceil ^{in} _{out}\quad .$$ P_R = (36)

From the above equations, we then obtain the form of the Hamiltonian for our system

$$H = P_R \dot{R} - L = - \frac{R}{G_N} \left( \beta _{in} - \beta _{out} - \kappa N R \right) \quad .$$ (37)

Since L is independent of $\dot{N}$, it follows that the corresponding conjugate momentum vanishes identically (eq.(35)). That relation represents a primary constraint which reflects the invariance of S _eff under any $\tau$-reparametrization which maps the boundary ($\tau _i$, $\tau _f$) into itself: for $\tau \rightarrow \tilde{\tau} = \tilde{\tau} (\tau)$ with $\tilde{\tau _i} = \tau _i$ $\tilde{\tau _f} = \tau _f$, we have $S _{eff} \rightarrow S _{eff}$, as it is easily seen if one keeps in mind the tensorial character of N ² = - g ₀₀. This primary constraint, in turn, generates a secondary constraint, namely the vanishing of the Hamiltonian (37). Indeed, the equations of motion of the shell are obtained by requiring that S _eff be stationary under variation of the functions $N(\tau)$ and $R(\tau )$ subject to the overall condition that they vanish at the (fixed) boundaries $\tau _i$ and $\tau _f$: the variation of S _eff gives

$$\delta S_{eff} = \int ^{\tau _f} _{\tau _i} \left\{ \left[ ... ...ial L}{\partial N} \delta N (\tau ) \right\} d \tau \quad .$$ (38)

Then, demanding that $\delta S_{eff}$ vanishes, yields two independent relations. The first

$$\frac{\partial L}{\partial N} = \frac{R}{G_N N} \left[ \bet... ...ta _{out} - \kappa N R \right] = - \frac{H}{N} = 0 \quad$$ (39)

is not a true equation of motion, since $\ddot{R}$ does not appear in it. Rather, Eq.(39) represents a constraint on the physically allowed states of our system, and since $N(\tau) \neq 0$, it implies the (weakly) vanishing of the Hamiltonian

$$H = 0 \quad.$$ (40)

On the other hand, the second equation following from the requirement that $\delta S_{eff}$ vanishes,

$$\frac{\partial L}{\partial R} - \frac{d}{d \tau} \left( \frac{\partial L}{\partial \dot{R}} \right) = 0$$ (41)

can be rearranged in the form

$$\frac{d}{d \tau} \left( \frac{H}{N}\right) = 0 \quad ,$$ (42)

which tells us, on account of eq.(40), that the constraint H=0 is preserved in ($\tau$) time, and that the lapse function is completely arbitrary, as expected. As a matter of fact, all classical and quantum evolution of the shell is encoded in the Hamiltonian constraint (40). However, the dynamics of the shell cannot be fully specified without fixing a gauge in our reparametrization invariant formulation, and this invariance is reflected in the arbitrariness of $N(\tau)$. A natural choice is the gauge $N \equiv 1$, which corresponds to selecting $\tau$ as proper time along the world history of the shell. In this gauge, the constraint reads

$$R \left( \beta _{in} - \beta _{out} \right) = 4 \kappa R ^2 \quad ,$$ (43)

which is just the matching equation [10], now playing the role of a Hamiltonian constraint describing the classical motion of the shell.